What does the semicolon ; mean in a function definition

A semicolon is used to separate variables from parameters. Quite often, the terms variables and parameters are used interchangeably, but with a semicolon the meaning is that we are defining a function of the parameters that returns a function of the variables.

For example, if I write $f(x1,x2,\ldots;p1,p2,\ldots)$ then I mean that by supplying the parameters $(p1, p2,\ldots)$, I create a new function whose arguments are $(x1, x2,\ldots)$.

So the general syntax is $functionname(variables;parameters)$.

In Noll's theorem it says that the function created by supplying $\partial \Omega$ is the same as that created by supplying $\vec{N}$. That's rather a nice way of saying that the function created only depends on $\vec{N}$.

There is no hard mathematical difference between the comma (,) and the semicolon(;).

The semicolon is used sometimes to optically separate some variable group. So the semicolon is not more than a reading aid.

The situation can be compared to the usage of different kind of parentheses, to make complex nestings more readable.